quantum entanglement: where dark energy and negative gravity … · 2013-12-24 · quantum particle...

Journal of Quantum Information Science, 2013, 3, 57-77 http://dx.doi.org/10.4236/jqis.2013.32011 Published Online June 2013 (http://www.scirp.org/journal/jqis)

Quantum Entanglement: Where Dark Energy and Negative Gravity plus Accelerated Expansion of the

Universe Comes from

Mohamed S. El Naschie Department of Physics, Alexandria University, Alexandria, Egypt

Email: [email protected]

Received March 11, 2013; revised April 27, 2013; accepted May 16, 2013

Copyright © 2013 Mohamed S. El Naschie. This is an open access article distributed under the Creative Commons Attribution Li- cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

Dark energy is shown to be the absolute value of the negative kinetic energy of the halo-like quantum wave modeled mathematically by the empty set in a five dimensional Kaluza-Klein (K-K) spacetime. Ordinary or position energy of the particle on the other hand is the dual of dark energy and is contained in the dynamic of the quantum particle modeled by the zero set in the same five dimensional K-K spacetime. The sum of both dark energy of the wave and the or-

dinary energy of the particle is exactly equal to the energy given by the well known formula of Einstein which is set in a four dimensional spacetime. Various interpretations of the results are presented and discussed based on

the three fundamental energy density equations developed. In particular

2E mc

5 2 2ordinary 2 22E mc mc where E

is the energy, m is the mass and c is the speed of light, 5 is Hardy’s quantum entanglement and 2 1 5

gives results in complete agreement with the cosmological measurements of WMAP and Supernova. On the other hand

2dark 21 22E mc gives an intuitive explanation of negative gravity and the observed increased rate of cosmic

expansion. Adding to one finds ordinaryE darkE 2 222 21 22E mc mc mc 2 which as we mentioned

above is Einstein’s famous relativity formula. We conclude that similar to the fact that the quantum wave interpreted generally as probability wave which is devoid of ordinary energy decides upon the location of a quantum particle, it also exerts a negative gravity effect on the cosmic scale of our clopen, i.e. closed and open universe. Analysis and con-clusions are framed in a reader friendly manner in Figures 1-14 with detailed commentary. Keywords: Dark Kinetic Energy of the Quantum Wave; Ordinary Position Energy of the Quantum Particle; Anti

Gravity; Negative Curvature; Collapse of the Hawking-Hartle Quantum Wave of the Universe; Revising Einstein’s Relativity; Quantum Gravity

1. Introduction

We utilize the wave-particle duality [1,2] and quantum entanglement [2-8] as fundamental features of quantum mechanics [1,8] as well as classical field, quantum set theory [3] and transfinite quantization [4-44] to develop an effective quantum gravity energy theory [1-8] that can convincingly resolve the problem of the accelerated rate of cosmic expansion and the missing dark energy of the cosmos and at the same time agrees with the accurate WMAP and Supernova cosmic energy density measure-ments and analysis [9-18]. It turns out that ordinary en-ergy is intimately related to the position energy of a ge-neric quantum particle [9,20] in five dimensional Kaluza-

Klein spacetime [2,4,6] and amounts to

5 2 21 2 2E mc mc 2

where E is the energy, m is the mass, c is the speed of light, 5 is Hardy’s quantum entanglement and

2 5 1 [12,17,18] (see Figure 1). Dark energy on the other hand was found to be a hitherto unknown and unsuspected halo-like feature of a generic quantum probability wave with a negative kinetic energy inducing negative gravity (see Figure 2). Thus a certain broken symmetry governs particles and waves from a set theoretical perspective [1,8]. In short the quantum particle is the ultimate source of ordinary energy and contributes to

Copyright © 2013 SciRes. JQIS

M. S. E. NASCHIE 58

(a)


M. S. E. NASCHIE 59

(b)

Figure 1. (a) The set theoretical ordinary energy-dark energy duality; (b) The quintessence of dark energy from the topology f the cantor set. o


M. S. E. NASCHIE


60

about 100 1 22 4.5 percent of the total energy of the cosmos [18,19] (see Figures 1-12). In turn the quan-tum particle is modeled by the zero set represented di-mensionally by von Neumann-Connes dimensional func-tion [8,44] of a Penrose quasicrystal fractal tiling space [21-23]. The broken mirror image of the quantum parti-cle is the halo-like quantum probability wave which is the source of a negative energy anti-gravity (see Figures 9 and 11) stemming from modeling the quantum wave via the empty set of the same Penrose quasicrystals frac-tal tiling [21-23,30-34,44]. Subsequently the energy of the quantum wave was found to be [12,18]

;OD (1b)

where 5 1 2 as in von Neumann-Connes’ non-commutative geometry [8,22,33] and its well known dimen dimensional function when applied to Penrose fractal clopen i.e. closed and open universe tiling [21,44].

2.2. Dark Energy, the Empty Set and the Quantum Wave

Dark energy on the other hand is given by the dual addi-tive hyper Hausdorff measure or volume of the empty set i.e. quantum wave in 5D Kaluza-Klein spacetime [4,6]. Thus we start from which is the empty set given by [8]

1D 2 2 22 5 2 21 22E mc mc

It was then found to the Author’s delight that the sum of ordinary energy of the particle and absolute value of the negative dark energy of the wave added together is exactly the energy predicted by Einstein’s formula [2,3, 17]

21 1,D (2)

i.e. [8,44]. 211 and forT HD D D

5 2 21 2

15

2E E E mc mc

2.

2.3. The Hausdorff “Volume” of the Zero Set in

Five Dimensional Space and the Quantum Particle

Our most important conclusion in this work is that Einstein’s gives the correct total energy for-mula for an infinitely large Cantor set-like clopen i.e. closed and open universe [24-29] but it does not distin-guish between ordinary energy and negative dark energy of a basically fractal cosmos. Noting that measurement leads to quantum wave collapse, it is completely normal that we cannot measure dark energy using present day technology. In the following sections we will attempt to explain more accurately but concisely the points trans-acted above. For a quick overview, the entire analysis and results are summarized in Figures 1-3, while valu-able details and additional information are presented in Figures 4-12. In Figures 13 and 14 we give a naive yet very instructive geometrical visualization and interpreta-tion of the wave-particle duality in five dimensional Kaluza-Klein spacetime.

2E mc For the hyper Hausdorff volume of the fractal zero set of ordinary energy [8], we apply the intersection rule of sets in 5 Kaluza-Klein dimensions and find (see Figures 1(a) and (b))

5Vol 5 Zero set .D (3)

In a sense this Hausdorff measure is the “volume” or the energy density of a quantum particle [8].

The volume 5 could be naively interpreted geomet-rically as the generalization of the volume of a cube to a pseudo 5D cube with each side being of a length equal to . (See Figures 13 and 14)

2.4. The Hausdorff “Volume” of the Empty Set in Five Dimensional Space and the Quantum Wave

On the other hand for the additive Hausdorff “volume” which is equal the length of the empty set [22,33] of dark energy we have the addition or union rule of sets in 5 K-K dimension and one finds (see Figures 1(a) and (b))

2. Basic Concept

2.1 Ordinary Energy, the Zero Set and the Quantum Particle

2 2 2 2 2 2

Vol 5 Empty set

5 .

D

(4)

Ordinary energy of the universe is directly proportional to the fractal nature of spacetime [4-12,30-34]. More accurately it is the multiplicative hyper Hausdorff meas-ure or volume of the zero set i.e. the quantum particle in a five dimensional Kaluza-Klein universe 5D . The zero set which possessrs the cardinality of the continuum is fixed using the Bi-dimensions formula [8]

We used here the union of sets rather than the intersec-tion because of the duality between particle and wave [2,8] (see Figures 13 and 14). Consequently the Haus-dorff measure of the empty set in K-K spacetime repre-sents the energy density of a quantum wave which does not exist in the usual conventional quantum interpretation because a conventional quantum wave in orthodox quantum mechanics is a probability wave and has no

,T HD D D (1a)

i.e. the topological dimension and the Hausdorff dimension

TD

HD so that we find

M. S. E. NASCHIE 61

Figure 3. Comparative analysis of ordinary quantum particle positive energy and dark quantum wave negative energy.


M. S. E. NASCHIE 62

Figure 4. The universe as a fractal in infinity space.


M. S. E. NASCHIE 63

Figure 5. Special relativity with extra dimensional twist or all roads lead to Rome and all theories lead to E mc2 22 .


M. S. E. NASCHIE 64

Figure 6. Yang-Mills for dark energy. Deriving E mc2 22 from quantum field theory in curved spacetime.


M. S. E. NASCHIE 65

Figure 7. A Beautiful mind for dark energy. A flow chart starting from Nash embedding and Witten’s M-Theory explaining the logical connections and sequential deduction using Einstein relativity and leading to quantum relativity energy E mc2 22 .


M. S. E. NASCHIE


66

and varying speed of Light Theory

P

mcE

mc

E

2

2

1

. Figure 8. Bosonic strings interpretation of E mc2 22

ordinary energy and collapses at measurement [1,2]. and collapse at measurement [1,2]. Again a possible naïve- geometrical interpretation of 25 is being a one dimen-sional “surface” i.e. the circumference of a pentagon with each of the 5 sides being of a length equal to 2 (see Figure 13). In particular the quantum wave determines the most probable location of a quantum particle but van-ishes for a classically large object which is the contrary effect of negative gravity induced by the same wave i.e. anti gravity vanishes locally and exists only by accumu-lation at cosmic scales.

2.5. The Average Hausdorff Volume of Quantum Particle and Wave as Total Relativity Energy

From the above points (3) and (4) we see that the total geometrical density or pseudo Hausdorff “volume” is the sum of both the fractal part of spacetime 5 and the

voids “empty” part of spacetime 25 . This means (see Figures 1(a) and (b) as well as Figures 13 and 14)

2Vol total 2 5 5 (5)

This is an extremely important result because an arithmetic average geometrical density or Hausdorff “dimensional volume” would give us

5 25 vol total 2 1. (6)

On the other hand the measure or “volume” or geo-metrical density of the unitarity set is 1 1;1D

5vol 1 1 1D and leads directly to Einstein’s celebrated energy density formula [2,5,17] (see Figures 1 and 2)

2 2vol 1 ,E mc D mc (7)

where E is the energy, m is the mass and c is the speed of light [2,3,12,17].

M. S. E. NASCHIE 67

(a) (b)

Figure 9. The deriving force behind gravity and negative gravity.

The fractal meaning of the constancy of the speed of light is explained in Figure 11 to round up the discussion and stress the fundamental role played by fractal geome-try of spacetime in determining the duality of ordinary and dark energy [12-14,22,33].

3. Analysis

3.1. The Speed of Light, Positive Energy, Negative Energy and Einstein’s Energy

Noting the result of point (2.5) we see that Einstein’s equation may be viewed as based on an arithmetic aver-age density or “volume” of 5D Kaluza-Klein space which does not differentiate between fractals due to voids and the voids in spacetime themselves and consequently Einstein’s energy density is simply given by the famous equation based on continuous geometry and unit hyper volume equal to one [12,26]:

2Einstein 1 .E mc 2mc (8)

On the other hand ignoring the density of the voids and considering only the fractal part of spacetime repre-sented by the five dimensional zero set, we then find the quantum relativity formula of ordinary energy density validated by direct cosmic measurement namely [12,17, 18] (see Figure 2)

5 2 2

2 2

2 22

22.18033989 22

Ordinary Energy

Energy of the quantum particle Positive Energy

QRE mc mc k

mc mc

E o

(9)

Note that division by 2 in the above equation is due to arithmetic averaging and not due to using Newton’s ki-netic energy because in this particular interpretation ve-locity tends to be the velocity of light without multiply-ing it with as in unit interval physics using Sigalotti’s critical value [28,29]. For an alternative derivation of

see Figures 3-8 as well as Figures 13 and 14. QREBy contrast if we consider spacetime voids only as

represented by the quantum wave i.e. the empty set in 5D Kaluza-Klein theory (which means that it is the so-called dark energy of the cosmos), then we have the following complementary quantum relativity energy density for-mula: (see Figure 2 as well as Figures 13 and 14)

2 2 2 2215 2 21 22

22

Dark Energy Energy of the quantum wave

Negative Energy Halo Energy

CQRE

kmc mc mc

k

(10)


M. S. E. NASCHIE 68

(a)

(b)

Figure 10. (a) Complexity of maximal entropy and minimum entropy are equal; (b) Negative dimension and negative Kelvin temperature.


M. S. E. NASCHIE 69

Figure 11. An overview regarding the constancy of the speed of light beeing a fractal mean value at the classical-quantum interface and thus constant. where

3 3 51 2 0.18033989 and 0.618033989.k

[22,34]

3.2. Einstein’s Energy Formula and a New Point of View

From points discussed in the preceding sections we can

conclude that Einstein’s formula is blind to the difference between ordinary ene ntum par-ticle energy and dark energy i.e. the quantum wave en-ergy because it presupposes a continuous smooth space-time and never addresses any questions related to fractal-ity or sparseness of spacetime which is the cause of the constancy of the speed of light at the classical interface being a fractal average (see Figure 11) [8,12,13,30]. However there are more profound consequences of the

2E mcrgy i.e. qua


M. S. E. NASCHIE 70

Figure 12. An overview r rding Gödel’s universe. ega


M. S. E. NASCHIE 71

D 21vol 5 Figure 13. A naive geometrical interpretation of geometrical density or Hausdorff measure which decide

upon the magnitude of dark energy of the quantum wave E mc2 21Dark 5

2 .

spacetime voids than that which will be discussed in the next paragraph. For the moment we have only to recall that the wave is the cobordism i.e. the surface of the quantum particle and thus particle and wave are insepa-rable [8].

3.3. The Negative Sign of the Quantum Wave Energy and Anti Gravity

If we consider the effect of the tw

sions of the zero set and the empty set on our average pseudo Hausdorff Kaluza-Klein hyper volume or geo-metrical density then we could work out the following modified negative arithmetic average [8] (see Figures 1 and 2)

o topological dimen-

5 21 5

Vol Total 5 22

o

(11)

nergy acquires in this case the Consequently dark e


M. S. E. NASCHIE 72

Figure 14. A naive geometrical interpretation of D 20vol

of the quaof the ordinary energy ntum particle

E mc2 21ordinary .

2

(12)

i-ly

the e rs

observations [9-17].

4. Intermediate Discussion We conclude t i.e.no

negative sign of the topological dimension of the empty set corresponding to the basic compactification equation of bosonic string theory [2,6,17]

4 26D 4 26 22 compactifiedD D

This negative sign (see Figure 9) is therefore an adtional indication that dark energy or what is equivalen

nergy of the quantum wave i.e

dt

. the empty set diffefundamentally from ordinary energy for being negative

and represents effectively negative gravity pushing in the opposite direction of classical gravity and causing the cosmic expansion of the Universe to accelerate rather than de-accelerate in agreement with relatively recent accurate

hat while fractals the zero set or phe-menologically the quantum particle is behind ordinary


M. S. E. NASCHIE 73

energy, voids in spacetime c rresponding to the empty set i.e. the quantum wave i.e. the halo of the quantum particle [8,23,33,44] are not only behind dark energy but also are the cause of negative gravity and the increased rate of cosmic expansion [9-11,15,16]. Negative gravity is essentially due to negative curvature of spacetime. If we see gravity as the Riemannian curvature of

o

4 4D related to

to the anticlastic havior of

polar me-tually simu-

it we have

hick direction then

direc-22 has

spacetime, then negative curvature [9] is and is akin ) known from the be

s and theories of elasticity and plasticity ke in Cosserat micro

theories may acilar to Cartan relativity and adm

antigravity [2]. Thus enological explanation arising

n’s ratio that when a long tn the longitudinal

opposite perpendicular , we could say that

the effect of an anti-curvature producing anti-gravity [9] due to negative energy of the halo [9] of the quantum particle i.e. the quantum wave which we have dubbed dark energy for want of a better physical word commu-nicating the deep meaning [9-18]. We can go even fur-ther still and claim that negative absolute temperature is another physical manifestation of negative dimensions and the empty set (see Figure 10) and therefore negative energy and consequently dark energy [41]. The episte-mological question of how a wave devoid of ordinary energy could induce a measurable negative gravity effect is essentially the other side of the fact that this same wave decides on the location of a quantum particle as confirmed by theory and experiment [1-3]. In other words the quantum probability wave possesses an onto-logical quality. Said differently we know that measure-ment collapses the quantum wave by converting an empty set particle to zero set wave. Consequently cosmic measurement collapses the Hawking-Hartle wave solu-tion of the Wheeler-DeWitt Schrödinger equation of the universe and therefore we cannot see or measure any energy of this wave unless we develop quantum nonde-

5. Various Interpretations and Duality in Five Dimensional Kaluza-Klein Spacetime

is equal to Einstein’s famous equation [2]. Unlike ordinary ewhich is entirely rooted in the zero set describingne

gies in the Magueijo-Smolin rmalism [5,26] and their elegant energy expression (see Figure 3):

compactified 22D curvature (see Figure 9elastic material[35-37]. A torsion tensor lidia and similar classical field late something simnegative curvature andhere the intuitive phenomfrom the effect of Poissobeam [35-37] is curved icross section curves in thetion [35]. Consequently D

molition measurement instrument [1].

Ordinary energy density and dark energy density are two dual yet complementary faces to a one common energy given by the sum of both which

2E mc nergy a ge-

ric quantum particle state, dark energy is solely due to the quantum wave aspect of the quantum particle as modeled by the empty set [8]. Nothing could show this symmetric duality better than writing both types of ener-

fo

2

2.

1P

mcE

mcE

(13)

Here m is the five dimensional Kaluza-Klein mass [38,39], c is the speed of light and PE is the Planck energy. Using the unit interval set theoretical value for

2Pmc E it was reasoned earlier on t e must have:

5hat w

2 25, and Pm c E that [12,17] (see Figure 3)

and inser E we find ting in

2 2

2 2 2

1 21 22

2 22 Ord

mc mcE

k k

mc mc E

inary

(14)

exactly as should be. However using Witten’s T-duality [2,4,6,40] (see Figure 3):

2

2T P

P

Emc

E mc (15)

and inserting in E one finds

22

2 2 2

21 221

121

5 2 21 22 Dark .

mcE mc k k

k

mc mc E

(16)

where 3 31 0.18033989k us we have demonstrated in the p

[22,33]. Th resent work and

presumably for the first time that dark energy is essen-tially the absolute value of the negative energy density of the Hawking-Hartle quantum wave solution of Feyn-man-Wheeler-DeWitt Schrödinger equation of the entire universe and as such it induces the hitherto unknown phenomena of negative gravity which we and others con-sider to be the cause of the observed increased rate of cosmic expansion [9-18]. This physical interpretation makes a great deal of sense because the et theoretical definition of the quantum wave is the e set repre-sent

smpty

ed by 2Dim empty set 1, whose topological dimension is uently the accurate way to write down the quantum wave energy density is [8,9,32]

minus one (−1). Conseq

2 2

2

5 2

21 22

E mc

mc

(17)

where 5 is the dimension of Kaluza-Klein spacetime [42, ding of the broken

ticle and wave and 43]. To gain an even deeper understansymmetry and duality between parconsequently between ordinary energy and negative or dark energy, let us start this time from Newton’s classical


M. S. E. NASCHIE 74

expression for the kinetic energy [2]

21.

2NE mv (18)

Recalling the set theoretical duality between the quan-tum particle zero set and the quantum wave empty set [8]

0 1 and 2 (19)

sformation given for particle and wave as in the following box:

Box No. 1. Particle-wave duality transformation.

For particle For wave

as well as the dual tran

3m m

v c

5m m

v c

Then inserting in NE one finds the following particle

and wave energy as per the next box: Box No.2. Quantum particle ordinary energy and quantum wave dark energy.

For particle For wave

3 2 2

5 2

2

1

2

2

22

Ordinary Positive Energy

NE E m c

mc

mc

2 2

2 2

2

15

2

5 2

21 22

Dark Energy

NE E m c

mc

mc

Let us contemplate the meaning of 3 in 3m and 5

in 5m as well as in c . The simplest way to explain it in the present context is in c . This stems from Sigalotti’s critical speed given by .v c The 3 on the other hand is 3 31 1 4n where

34n is the Hausdorff dimension of a four di-mensional set given by 34;4 and representing the core of fractal-Cantorian spacetime [21,22]. At the same time 3 is the global quantum entanglement probability or what some call the counter factual effect of the space-time surrounding a quantum particle [32]. It is numeri-cally equal to the Unruh dimensionless thermal tempera-ture [2] and together with 2 gives 3 2 5

the which is glement of two quantum particles [37]. Oner hand the appearance of 5 in 5m maybe seen as the

oth

gnetism

five dimensionality of a Kaluza-Klein spacetime [2,4,6]. This is an ordinary 4D-Einstien space with a single addi-tional cyclical pipe-like compactified extra dimension joining electro ma to gravity [38,39,42,43] (see Figure 4). It could be also seen as an extra spin half fer-mionic dimension [22]. However like the bosonic fractal dimension 34 enters into the energy equation of the quantum particle in the inverse form 3 31 4 , the non-fractal five dimensional fermionic K-K spacetime

[38,39] enters into the energy equation of the quantum wave or dark energy directly simply as factor 5. For the sake of completeness let us see what would result from considering a fractal i.e. a transfinitely quKaluza spacetime developed by the Author [38,39, 42,43] with

antized Klein-

35 5 which means (see Figure 4)

3 15 5 .

14

14

4

(20)

In this case 3 25 would mislead us into wron ergy is equal to Einstein’s

1 and 4) gly thinking

energy (see Figures

that dark en

3 2 2

2

15

2

Energy

Particle Position Energy W

2 1 2 Total Potential Energy

E mc

mc

(21)

This is initially a surprising result but it is a misunder-standing as well because it shows that E(Einstein) tainable from a fractal five-dimension Kaluza-Klein spacetime while dark energy is the child of a mere five-

ional non-fractal Kaluza-Kl me. How-

2 Einsteinmc E

2 2 3 2 25 2 2

Ordinary Energy Dark

mc mc

ave kinetic Energy

is ob-

dimens ein spacetiev 3er the difference is namely that is the factor re-sponsible for the additional energy which is nothing but the ordinary energy which when added to non-fractal 5D Klein Kaluza [38,39] i.e. to dark energy gives the total energy, namely that of Einstein’s energy. This confirms all of our earlier results namely that 2E mc could be obtained via either a smooth 4D spacetime or via a deli-cate analysis using a fractal 35D five dimension spacetime (see Figure 4). Seen that way 35D which is related to the fractal 11-D theory via

511D is a more fundamental theory than Ein-stein’s D 4 spacetime t eh fact we could de-duce 35D

ory. In from 511D by dividing by 2

and subtracting the number weight of 1 photon [20] namely so that one finds (see Figure 4)

5 311 2 5 (22)

In other words we can either interpret 5 in hyper volume or Hausdorff measure an visio 2 as

1En by

as a d di

averaging or we see it as the transformation 3m m and v c and division by 2 as coming from using Newton’s kinetic energy.

We conclude this section by stressing that there are three and not only one Kaluza-Klein theories. The first is


M. S. E. NASCHIE 75

the classical 5D theory of Kaluza [41]. The second is the quantized 5D theory of Klein [41,42]. The third is the transfinitely quantized theory dthe author [38,39,43].

Dark energy is a non-classica of neg

35 D eveloped by

6. Conclusions

l form ative energy density stemming from the quantum wave empty set characteristic and we found it quantitatively to be given y the absolute value of b

2

2 2 dark 5 21 22

2E mc mc

(23)

where 2

nsion

could be rpreted as the local Ha type quantum entanglement of two particles or the Hausdorff ime of the empty set and

inte rdy’s

d 2 5 1 . Ordinary energy o the other hand is given by the

positive energy of the zero set descrn

ibing the quantum article and found in the present work to be equal [12] p

5 2 2ordinary 2 22E mc mc (24)

where 5 is the famous golden mean quantum entang nt [2,31].

There are numerous other methods and theories which lead to essentially the sam nclusion regarding i-nary positive energy density being equal to Einstein’s energy divided by 22 which are summarized in Figures

8. In particular Figure 7 shows how to derive

Hardy’sleme

e co ord

5-5 2 1 22 from Nash’s embedding us the n tion of

spacetime sparseness and Figure 8 gives an idea about the role of bosonic strings dimension.

A particular tantalizing result is that the sum of both energies, dark energy and ordina s a nea con-firmation of Einstein’s famous relativity formula because

pl o

ry energy, i t

5c

rvature as per

5 2 2

2 2

5 2 2

1 Einstein .

E m mc

mc mc E

(25)

The negatives of the dark energy on the other hand confirm earlier string theoretical conjectures that negative gravity is due to negative energy of the compactified 22 dimensions which produce negative cu the bosonic string dimensional equation [2,4,12]

4 26 224 22 22.D D D (26)

Following the same reasoning one can write the 21 22 factor of dark energy as

5 26

4 26

5 22 2121 22

4 22 22

D D

D D

(27)

which proves that 1 22 of ordinary energy and 21 22 of dark energy have opposite vent. In fact the present theory strongly suggests the physical

reality of Kaluza-Klein extra fifth dimension [42,43] as well as the fractality of spacetime and ontological reality of the quantum wave [8,13,14,20,43,44] all apart from the reality of dark energy which we cannot see or meas-ure because measurement collapses the quantum wave.

For nt view of negative gravity v

signs in any e

a differe ia general relativity and comparison with Heterotic string theory see Figure 9 and commentary therein.

It is worth noting that negative dimensions are behind the experimentally observed negative Kelvin absolute temperature [41] and may thus be looked upon as another ph tion of dark

he present work suggests, then our cosmos is clopen i.e. opened and closed because Cantor sets are clopen.

We could understand the repelling effect of dark en-ergy on a cosmic scale as a phenomenon basically due to the fact that the motion of quantum objects and its spatial location is determined completely by the quantumfollowing prescription [1-3]. It is a matter of speculation that the spinning of Gödel’s universe (see Figure 12) [40] is related to dark energy and repelling gravity. Finally the constancy of the speed of light as a fractal average is ex

ysical interpreta energy (see Figures 10(a) and (b)). Furthermore if the building blocks of spacetime are random Cantor sets as t

wave

plained in some detail in Figure 11.

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